Image Processing Reference
In-Depth Information
shows that the eigenvectors of
OO
T
are related to those of
O
T
O
through a pro-
jection. Accordingly, given
O
=[
f
1
,
···
,
f
K
], with
f
k
∈
E
M
, the eigenvectors of
OO
T
can be obtained from those of
O
T
O
as follows:
1. Compute the sorted eigenvalues and eigenvectors of
O
T
O
as
λ
m
,
ψ
m
. The
eigenvalues are also the eigenvalues of
OO
T
.
2. Obtain the (unnormalized) eigenvectors
ψ
, (15.34).
ψ
m
as
O
M
matrix
OO
T
has
M
eigenvalues that are
the same as the
K
eigenvalues of the
K
Exercise 15.1.
How is it that the
M
×
K
matrix
O
T
O
, and yet
K
×
M
?
Example 15.1. Insomefacerecognition techniques, recognition isdone bycompar-
ingthegrayimages,interpretedas(very)highdimensionalvectorsinaHilbertspace
withthescalarproduct,Eq.(3.27).Asimilarityordissimilaritymeasurebetweentwo
faceimagescanthenbeused,e.g.thedirectionaldifference,(3.56),ortheEuclidean
distance, to decide whether or not the images represent the same person. Alterna-
tively, a trained classifier, such as a neural network [29,49], or the Mahalanobis
distance [62], can be used in this decision making. One assumes then the reference
images of the clients,
O
=[
f
1
,
E
M
. In order for this to work,
the images must be scale- and position-normalized, so that little or no background
is present and the eyes are essentially in the same position in all images. It has been
shown, see [202,220] and others, that this recognition is improved if the dimension
ofthefaceimagesisfirstreducedto
N
byPCA,wheretypically
N<K M
.The
resulting subspace is also called the face space. The similarity of two face images
are then measured in this subspace, spanned by the
N
most significant eigenvectors,
also called eigenfaces.
In the top row of Fig. 15.1, which shows six images of two persons, some sam-
ples of a face training set
O
are illustrated. Representing 64
···
,
f
K
] with
f
k
∈
96 images, the cor-
responding face vectors have
M
= 6144 dimensions. The training set
O
consists
of
K
= 738 such face vectors. The mean of the training set is subtracted from the
training set, and also later from the test set, (15.22). The actual scatter matrix
OO
T
is 6144
×
6144, whereas the alternative scatter matrix
O
T
O
, by which the eigen-
vectors can be computed, is 738
×
738. The 24 most significant eigenvectors are
shown as pseudo colored images in the same figure, in reading order. Using a test
set containing 369 images (different than those in
O
) and retaining the
N
=30
most significant eigenvectors, one could obtain 89% recognition in the senspe that
the person to be recognized was assigned the top rank. The recognition was 96% if
the correct image was found among the top 3 ranks [154].
×
